Coulomb-excitation of 112, 114, 116Sn Pieter Doornenbal

The three faces of the shell model Pairing interaction: large spin-orbit splitting implies a jj coupling scheme.

Symmetries of the shell model Three bench-mark solutions: No residual interaction  IP shell model. Pairing (in jj coupling)  Racah’s SU(2). Quadrupole (in LS coupling)  Elliott’s SU(3). Symmetry triangle:

Racah’s SU(2) pairing model Assume large spin-orbit splitting ls which implies a jj coupling scheme. Assume pairing interaction in a single-j shell: Spectrum of 210Pb:

Solution of pairing hamiltonian Analytic solution of pairing hamiltonian for identical nucleons in a single-j shell: Seniority  (number of nucleons not in pairs coupled to J=0) is a good quantum number. Correlated ground-state solution (cfr. super-fluidity in solid-state physics). G. Racah, Phys. Rev. 63 (1943) 367

Superfluidity in semi-magic nuclei Even-even nuclei: Ground state has =0. First-excited state has =2. Pairing produces constant energy gap: Example of Sn nuclei:

Seniority scheme in Sn isotopes: 2 6+  = 0 2 4+  = 0 2 2+  = 2 0+ energy axis  jn J 6+ min   4+ j j j 2+ J j j j j J j J 0+ -residual interaction gives nice simple geometric rationale for Seniority Isomers from E(j2J) ~ -V0Frtan(/2) for T=1, even J

Example of the 8+ ( p1h9/2)2 isomers in nuclei with Z > 82 Seniority Scheme Seniority conserving Dn = 0 1-body even tensor B(E2: I  I – 2, I  2) Seniority changing Dn  0 B(E2: I  I – 2, I = 2) 2 8+ 2 6+ 2 4+ 2 2+ j = (9/2)n n = 0 0+ B(E2) Fractional Filling

e.g. Jp = (h9/2)2 coupled to 0+, 2+, 4+, 6+ and 8+. q q d-interaction gives nice simple geometric rationale for Seniority Isomers from DE ~ -VoFr tan (q/2) for T=1, even J 8 6 4 DE(j2J) 2 e.g. Jp = (h9/2)2 coupled to 0+, 2+, 4+, 6+ and 8+. 180 90 q 2 4 6 8 q

d-interaction gives nice simple geometric rationale for Seniority Isomers from DE ~ -VoFr tan (q/2) for T=1, even J 2 4 6 8

Reduced transition probability in a single J-shell ≈Nparticles*Nholes (2j+1) ≡ nucleons/orbital

Reduzierte Übergangswahrscheinlichkeit in einer einzelnen J-Schale ≈NTeilchen*NLöcher (2j+1) ≡ Anzahl der Nukleonen . in der j-ten Schale

Reduced transition probability in a complex shell ≈Nparticles*Nholes number of nucleons between shell closures.

Reduzierte Übergangswahrscheinlichkeit in einer komplexen Schale ≈NTeilchen*NLöcher Anzahl der Nukleonen zwischen den Schalenabschlüssen

Proton np-nh core excitations (t=n) Theoretical interpretation theory (neutron valence + proton core excitations and 90Zr as closed-shell core) theory (neutron valence and 100Sn as closed-shell core) Neutron/proton single-particle states in a nuclear shell-model potential: from A. Banu et al., cond. publ. t=0 t=2 t=4 Neutron number B(E2 ) e2 b2 This work •••••••• Proton np-nh core excitations (t=n) & 100Sn core is open

Previous measurements: 112Sn 2+ 112Sn 1257 320 (20) fs ≙ 0.244 (13) e2b2 75Gr30 0.229 (5) α, 16O Coul Ex 81Ba05 0.256 (6) 16O 57Al43 0.180 (40) α 61An07 0.33 (6) 14N, 20Ne 70St20 Reference* Measured Value Projectile Method 0+ 2+ 114Sn 1300 300 (60) fs ≙ 0.25 (5) e2b2 114Sn 0+ Coul Ex α 0.20 (7) 57Al43 14N, 20Ne 0.25 (6) 61An07 Coulex 16O 0.25 (5) 81Ba05 DSA 112Cd(α,2n) 0.238 (77) 91VIZW RDDS 100Mo(18O, 4n) 0.189 (39) 01Ga52 2+ 116Sn 1294 374 (10) fs ≙ 0.209 (5) e2b2 0+ *From NNDC

Coulomb excitation experiment 112,114,116Sn→58Ni at 3.6MeV/u Ex=1257MeV, 1300MeV, 1294MeV B(E2)↑=0.244(13), 0.25(5), 0.209(5)e2b2 Sn-excitation ~ 180 mb Ni-excitation ~115 mb γ-efficiency = 0.005 beam intensity = 1pnA target thickness = 1mg/cm2 10 % duty factor pγ-rate (Sn) = 1/s

Choosing the right target 2+ 1454 154 keV 2+ 184W → 120Sn @ 4.7 MeV/u 114Sn 1300 0+ 0+ 58Ni 184W 1120 1250 2+ 120Sn 1171 120Sn 2+ 0+ θγ = 25º 4+

Important to know: Transition Ratio 90º-140º 2+→0+ 1 4+→2+ 0.017 116Sn θγ = 25º 116Sn→58Ni Secure energy:

Conclusion: Very easy to perform, yet leads to interesting physical results All necessary equipment is already available at GSI. Only feasible using Sn ion beams We ask for a total of 3 times 7 shifts of beam time for the isotopes 112, 114, 116Sn.